Quasiclassical and semiclassical wave-packet dynamics in periodic potentials

Balzer, Birgit; Dilthey, Stefan; Stock, Gerhard; Thoss, Michael

doi:10.1063/1.1601219

Cited by 12 publications

(5 citation statements)

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“…The analysis leading to the minimum uncertainty delivers a lowering operator, and this suggests that one solve eq III.1 by a factorization approach. In the supersymmetry approach, we expect to factor the Schro ¨dinger equation in the form 11,12 where W(φ) is the superpotential, in terms of which 7,8 As is well-known, eq III.5 is the Riccati substitution for the Schro ¨dinger equation, and in this case, we immediately identify W(φ) as (compare eq III.7 with eq III.4) eq III.7 with eq III. 4.…”

Section: Supersymmetric Quantum Dynamics Of the Hindered Rotormentioning

confidence: 99%

“…The analysis leading to the minimum uncertainty delivers a lowering operator, and this suggests that one solve eq by a factorization approach. In the supersymmetry approach, we expect to factor the Schrödinger equation in the form , true[ − normald normald normalφ + W ( φ ) true] true[ d d φ + W ( φ ) true] ψ = E ψ where W (φ) is the superpotential, in terms of which , V ± ( φ ) = W 2 ( φ ) ∓ d d φ W ( φ ) …”

Section: Supersymmetric Quantum Dynamics Of the Hindered Rotormentioning

confidence: 99%

“…1 It also gives a hint of which quantum states are most compatible with classical mechanics. [2][3][4][5][6][7][8][9] In these states, the uncertainty product of the kinematic variables equals the expectation value of the imaginary part of their commutator. For the case of position and momentum operators x ˆand p ˆ(or when setting p ≡ 1, x ˆ, and the wavenumber operator, k ˆ), these operators, along with the identity, I ˆ, observe the so-called canonical commutation relations associated with the Heisenberg-Weyl Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

“…The Heisenberg uncertainty principle is fundamental to quantum mechanics since it precludes the existence of precise simultaneous measurements of position and momentum . It also gives a hint of which quantum states are most compatible with classical mechanics. − In these states, the uncertainty product of the kinematic variables equals the expectation value of the imaginary part of their commutator. For the case of position and momentum operators x̂ and p̂ (or when setting ℏ ≡ 1, x̂ , and the wavenumber operator, k̂ ), these operators, along with the identity, Î , observe the so-called canonical commutation relations associated with the Heisenberg−Weyl Lie algebra. ,, The uncertainty minimizers for this case are the “canonical” coherent states which result from requiring that the uncertainty product of the position and momentum variances to attain the absolute minimum.…”

Section: Introductionmentioning

confidence: 99%

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The Heisenberg−Weyl Algebra on the Circle and a Related Quantum Mechanical Model for Hindered Rotation

et al. 2009

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We discuss a periodic variant of the Heisenberg-Weyl algebra, associated with the group of translations and modulations on the circle. Our study of uncertainty minimizers leads to a periodic version of canonical coherent states. Unlike the canonical, Cartesian case, there are states for which the uncertainty product associated with the generators of the algebra vanishes. Next, we explore the supersymmetric (SUSY) quantum mechanical setting for the uncertainty-minimizing states and interpret them as leading to a family of "hindered rotors". Finally, we present a standard quantum mechanical treatment of one of these hindered rotor systems, including numerically generated eigenstates and energies.

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Section: Supersymmetric Quantum Dynamics Of the Hindered Rotormentioning

confidence: 99%

Section: Supersymmetric Quantum Dynamics Of the Hindered Rotormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Heisenberg−Weyl Algebra on the Circle and a Related Quantum Mechanical Model for Hindered Rotation

et al. 2009

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“…Probably, the transition between both can be best scrutinized by exploiting phase-space methods [58][59][60]. This opens up the possibility of gaining some information about the nonclassical behavior with a quasiclassical description that employs essentially classical trajectories, while the correct quantum initial state is taken into account via, e.g., the Wigner function [61][62][63]. Despite some problems with the interpretation, the Wigner function has enjoyed substantial attention in various domains of physics [64] and has already been applied to some nonlinear problems in quantum optics [65][66][67][68][69].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear cross-Kerr quasiclassical dynamics

et al. 2013

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We study the quasiclassical dynamics of the cross-Kerr effect. In this approximation, the typical periodical revivals of the decorrelation between the two polarization modes disappear and remain entangled. By mapping the dynamics onto the Poincaré space, we find simple conditions for polarization squeezing. When dissipation is taken into account, the shape of the states in such a space is not considerably modified, but their size is reduced.The optical Kerr effect refers to the intensity-dependent phase shift that a light field experiences during its propagation through a third-order nonlinear medium. This leads to a remarkable non-Gaussian operation that has sparked considerable interest due to potential applications in a variety of areas, such as quantum non-demolition measurements [1-9], generation of quantum superpositions [10][11][12][13][14][15][16][17][18][19], quantum teleportation [20][21][22], quantum logic [23-28] and single-particle detectors [29][30][31], to cite only a few. Enhanced Kerr nonlinearities have been observed in electromagnetically induced transparency [32-35], Bose-Einstein condensates [36], cold atoms [37] and Josephson junctions [38-40]. Additional arrangements involve the Purcell effect [41], Rydberg atoms [42], light-induced Stark shifts [43] and nanomechanical resonators [44].Special mention must be made of the role this cubic nonlinearity has played in the generation of squeezed light. The first proposals employed schemes involving a nonlinear interferometer [45] or degenerate four-wave mixing [46,47]. But quite soon optical fibers became the paradigm for that purpose [48][49][50][51][52][53]. However, due to the typically small values of the nonlinearity in silica glass [54], Kerr-based fibers need long propagation distances and high powers, which bring other unwanted effects [55,56].In this paper, we direct out attention on this limit of high intensities, in which one could expect a classical description to be pertinent. Under reasonable assumptions, Maxwell's equations lead to a set of coupled nonlinear Schrödinger equations that has long been a useful tool for depicting the behavior of optical fields in nonlinear dispersive media. It has proved valuable in the description of such diverse phenomena as pulse compression, dark soliton formation and self-focusing of ultrashort pulses [57]. However, there remain non-classical features that cannot be explained in this classical manner. To put it differently, at the most basic level, the propagation of light in a Kerr medium is accompanied by unavoidable quantum effects.The considerations so far indicate that the regime we wish to explore can be regarded as a problem at the boundary between classical and quantum worlds. Probably, the transition between both can be best scrutinized by exploiting phase-space methods [58][59][60]. This opens New Journal of Physics 15 (2013) 043038 (http://www.njp.org/)

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The Classical Special Functions

The Mathematica GuideBook for Symbolics

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Quasiclassical and semiclassical wave-packet dynamics in periodic potentials

Cited by 12 publications

References 50 publications

The Heisenberg−Weyl Algebra on the Circle and a Related Quantum Mechanical Model for Hindered Rotation

The Heisenberg−Weyl Algebra on the Circle and a Related Quantum Mechanical Model for Hindered Rotation

Nonlinear cross-Kerr quasiclassical dynamics

The Classical Special Functions

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